In Theorem REF , the edge mode count via transversal crossings, \(n(\mathcal {C}_R, \mu )\) , is well-defined due to regularity properties of the edge modes branches for the unperturbed \(H_{R,\theta }\) , a family of order 2 ordinary differential operators with constant coefficients. For example, at each \(\theta \) , there are only finitely many edge modes, so they do not accumulate into the bulk spectrum. Moreover, as \(\theta \)  is increased, the number of merging points with the bulk spectrum, used in Def. REF  later, is also finite.
Such properties may not persist if we perturb \(H_{R,\theta }\) . It is therefore worth pointing out that \(n(\mathcal {C}_R, \mu )\)  is, up to a minus sign, a special case of spectral flow a la Phillips [1]}, [2]}, and that the latter still applies in the perturbed setting where the aforementioned pathologies may occur. We briefly recall its definition.
